Opacity dependence of elliptic flow in kinetic theory

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https://doi.org/10.1140/epjc/s10052-019-7262-x Letter

Opacity dependence of elliptic flow in kinetic theory

Aleksi Kurkela^1,2, Urs Achim Wiedemann^1,a, Bin Wu¹

1Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland

2Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway

Received: 21 May 2019 / Accepted: 1 September 2019 / Published online: 12 September 2019

Abstract The observation of large azimuthal anisotropies vnin the particle spectra of proton–proton (pp) and proton–

nucleus (pA) collisions challenges fluid dynamic interpreta- tions ofvn, as it remains unclear how small collision systems can hydrodynamize and to what extent hydrodynamization is needed to build upvn. Here, we study in a simple kinetic theory how the same physics that leads to hydrodynamization in large systems represents itself in small systems. We observe that one third to one half of the elliptic flow signal seen in fully hydrodynamized systems can be built up in collisions that extend over only one mean free pathlmfpand that do not hydrodynamize. This is qualitatively in line with observing a sizeablev2inppcollisions for which other characteristics of soft multi-particle production seem well-described in a free- streaming picture. We further expose a significant system size dependence in the accuracy of hybrid approaches that match kinetic theory to viscous fluid dynamics. The implications of these findings for a reliable extraction of shear viscosity are discussed.

1 Introduction

Ultra-relativistic nucleus–nucleus (AA), proton–nucleus (pA) and proton–proton (pp) collisions display remarkably large signatures of collectivity, in particular in the hadronic transverse momentum spectra and their azimuthal asymme- triesvn[1–6]. To infer the properties of the ultra-dense and strongly expanding QCD matter in the collision region from these data, a dynamical modelling of collectivity is indis- pensable. From comparing fluid dynamic models to data of large (AA) collision systems, one generally determines matter properties consistent with a perfect fluid that exhibits minimal dissipation (having minimal shear viscosity over entropy ratio,η/s) [7,8]. In marked contrast, the standard implementation of soft multi-particle production in general-

ae-mail:[email protected]

purpose event generators [9] of pp collisions implements a free-streaming picture according to which outgoing quanta do not interact with each other. Kinetic transport theory is of particular interest in this context since it can in princi- ple interpolate between the limiting cases of free-streaming and fluid-dynamic behavior. Indeed, transport models have been demonstrated to account for the signals of collectivity in pA and AA collisions with material properties that allow for a significant mean free path, thus exhibiting non-minimal dissipation [10–18].

Despite these recent advances in applying transport theory to hadronic collisions, our dynamical understanding of the system size dependence of collectivity remains incom- plete. Even elementary questions – such as: what is the minimal (maximal) size over which a hadronic collision system needs to extend to exhibit fluid-like properties (free streaming)? – still await systematic exploration. To address these questions, the need for developing even more realistic and more complex simulation tools of pp, pA and AA collisions is widely acknowledged. But to better understand generic, model-unspecific characteristics of how collectivity is built up in collision systems of sizeable mean free path, one should alsoexplore the opposite direction: One should also study in isolation particularly simple formulations of kinetic theory that are not embedded in the multi-layered reality of fully realistic simulation packages. Such formulations ideally depend on as few model parameters as possible, and they are ideally free of model-specific assumptions about, e.g., hadronization or about detailed dynamical approxima- tions entering the collision kernel.

Many microscopic models with boost-invariant longitudinal dynamics satisfy hydrodynamic constitutive equations in situations significantly out of equilibrium, an observation dubbed “hydrodynamization without thermalization”

[19–25]. Ultra-relativistic pp, pA and AA collisions realize such out-of-equilibrium scenarios since they are initiated with a highly anisotropic momentum distribution. However, whether the process of hydrodynamization is completed or

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not may depend significantly on the transverse extent and lifetime of the collision system. Here, we analyze system- atically over the entire range of physically relevant system sizes how a particularly simple, one-parameter kinetic theory can account for the requirements of realizing a close- to-hydrodynamic behavior on time scales comparable with a nuclear radius while supporting a close-to free-streaming picture in minimum bias pp collisions, and exhibiting for small but increasing system sizes a rapid onset of sizeable signals of collectivity. The simple kinetic theory employed here is based only on arguably generic assumptions about the isotropizing character of rescattering phenomena, and it will be shown to exhibit with increasing system size an increasing degree of hydrodynamization. The model thus provides a simple testbed for understanding to what extent signals of collectivity can arise in small systems with negligible or partial hydrodynamization, and how they can grow with increasing system size. In addition, this approach allows one to quan- tify the system-size dependent uncertainties that arise from interfacing a pre-hydrodynamic evolution based on kinetic transport with a subsequent fluid dynamic description.

In general, any kinetic transport formulation assumes a scale separation between the typical size of the wave-packet of particle-like excitations and their mean free path. This assumption is not realized, e.g., in models of strongly coupled liquids formulated in the limit of strong coupling with the gauge/gravity conjecture. This assumption would be sup- ported, however, by any evidence for the dominance of free- streaming in a small collision system, since free-streaming over some finite extent translates trivially to a lower bound on the mean free path. Moreover, any finite mean free path implies non-minimal dissipative material properties and thus translates to a non-minimal constraint onη/s. In this sense, establishing a unified dynamical description of collectivity valid from AA via pA to the smallest pp collision systems has the potential of providing a complementary constraint on η/sfrom the system size dependence ofvndata.

2 Kinetic transport: the model

Our study focusses on azimuthal asymmetriesvnof the transverse energyd E_⊥ that are trivially obtained from those of measured particle spectrad N,

d E_⊥ dηsdφ≡

d p_⊥² p_⊥d N d p²_⊥dηsdφ

= d E_⊥ 2πdηs

1+2^∞

n=1

vncos(nφ)

. (1)

In comparison tod N, an analysis of d E_⊥ is not compli- cated by the potentially confounding effects of hadronization. We calculated E_⊥by evolving the energy-momentum tensorT^μνof the system to late times. To this end, we write

T^μν = ₁

−1 dvz

2

_d_φ

2πv^μv^νF in terms of the first momentum momentF(x_⊥, , τ)= ₄_π_p²_{d p}

(2π)³ p f of the distribution function f. Here, pis the modulus of the three-momentum, and we use normalized momentav_μ≡ p_μ/pwithp_μp^μ= 0 andv⁰=1. The two-dimensional angular orientationof the momentum can be written in terms of the azimuthal angle φand the normalized longitudinal momentum componentvz. For massless boost-invariant kinetic transport in the slice of central spatial rapidityηs =0, the evolution equation forF reads [17]

∂τF+ v⊥·∂x_⊥F−vz

τ (1−v²z)∂vzF+4v²z

τ F= −C[F].

(2) For the collision kernelC[F], we use the isotropization-time approximation (ITA)

−C[F] = −γ ε¹^/⁴(x)[−vμu^μ](F−Fiso), (3) where ε is the local energy density and Fi so(x_⊥, , τ) =

ε(x_⊥,τ)

(−u_μv^μ)⁴ is the isotropic distribution in the local rest frame u^μ given by the Landau matching condition, u^μT_μ^ν =

−εu^ν. The ITA is closely related to the relaxation time approximation. We emphasize, however, that for observables constructed from T^μν, it is not necessary to spec- ify the momentum-dependence of C[F]. Equation (3) is solely based on the mild assumption that any system evolves towards an isotropic distribution and that this can be characterized for p-integrated quantities by a single isotropiza- tion timelmfp ∼

γ ε¹^/⁴₋1

, set by the only model parame- terγ. The ITA has been studied extensively in the hydro- dynamical limit and its transport coefficients are known:

τ_π = (γ ε¹^/⁴)⁻¹ and kinetic shear viscosity _sT^η = _{γ ε}¹1/41 5. While our calculations depend only on the combination _sT^η and do not depend on^η_s independently, the latter can be deter- mined once the equation of state relating energy density and temperature is specified. (If one choosesε≈13T⁴, as moti- vated by lattice results, one findsη/s ≈0.11/γ.) We note that the thermal equilibrium distribution that enters the relaxation time approximation (RTA) is a special choice of an isotropic distribution. Therefore, the RTA and ITA dynamics for T^μν are identical while the ITA does not assume relaxation to a local thermal equilibrium. The ITA is found to reproduce theT^μν evolution of the QCD weak coupling effective kinetic theory [26] within∼15% [25]. However, the following does not assume that the collision kernel is dominated by perturbative physics.

Azimuthal asymmetries vn in the final momentum distributions arise from azimuthal eccentricitiesn in the initial spatial distribution. To choose a longitudinally boost- invariant initial condition that shares pertinent phenomeno- logically relevant features, we assume at each point in space an azimuthally isotropic momentum distribution with max-

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0 0.5 1 1.5 2 τ/R

0 0.02 0.04 0.06 0.08 0.1

τ^4/3ε/3

τ^4/3P_L^Hydro(ε) τ^4/3P_L

γ^

= 1.68

0 0.5 1 1.5 2

τ/R 0

0.02 0.04 0.06 0.08 0.1

τ^4/3ε/3

τ^4/3P_L^Hydro(ε) τ^4/3P_L

γ^

= 3.36

0 0.5 1 1.5 2

τ/R 0

0.02 0.04 0.06 0.08 0.1

τ^4/3ε/3

τ^4/3P_L^Hydro(ε)

τ^4/3P_L γ^{^} = 8.41

Fig. 1 Time evolution of the energy densityτ^4/3ε(black) and the longitudinal pressureτ^4/3P_L(red), measured atr=0 and compared to the 1st order hydrodynamic constitutive equation (green) for different values of transverse system sizeγˆ

imal anisotropy in the longitudinal component (∝ δ(vz)).

For spatial distributions, we choose an azimuthally isotropic Gaussian density profile distorted by eccentricities n. Focussing for simplicity on the second harmonic, we write

F(x_⊥, , τ0)

=2ε0δ(vz)exp −r²

R² 1−2

r² R²cos 2θ

, (4)

with spatial azimuthal angleθand radial coordinater. The normalization of (4) corresponds to an initial central energy densityε(τ0,r = 0) = ε0. We takeτ0 → 0 keepingε0τ0

fixed. Then, evolving this initial condition (4) with the kinetic theory (2), dimensionless observables can depend only on opacityγˆ = R³^/⁴γ (ε0τ0)¹^/⁴. This opacity may be thought of as measuring the transverse system sizeRin units of mean free path at the timeτ = Rat which collectivity is built up,

ˆ

γ = R/lmfp(τ = R)≈ Rγ (eε(τ =R,r=0))¹^/⁴, where the latter equivalence is exact for a free streaming system.¹ From previous studies of this kinetic theory to first order inγˆ, i.e., for small system sizes, we know already that all linear and non-linear structures observed in the azimuthal anisotropiesvnarise, and thatv2/2=0.212γˆ[17].

Note that in physical collision systems, the opacityγˆcan be varied either by changing the geometrical size of the system R or by changing the mean free path by varying the densityε0τ0. In central and semi-central heavy ion collisions the geometrical system size can be controlled by selection of centrality classes. In contrast, in pp collisions, one expects that the change in geometrical size plays a lesser role butγˆ may still be varied by multiplicity selection leading to denser or more dilute systems.

3 Kinetic transport: non-perturbative solution and results

The first result reported here is that we have developed a novel approach for solving the kinetic theory (2) exactly to

1Here the Euler’s constantearises from the time evolution of the central density in a free-streaming systemτε(x_⊥=0, τ)=τ0ε0e⁻^t^τ²^/^R²+ O(γ, 2).

all orders inγˆ and thus for collision systems of any opacity.

We do so by discretizing the transport Eq. (2) in comoving coordinates that leave the distribution F˜ of free streaming particles unchanged as a function of time, and evolve it in time numerically

∂_τF˜n(x˜_⊥, φ,v˜z, τ)= −e⁻^{i n}^(φ−θ) ⁴ Cn

⁴e^{i n}^(φ−θ)F˜n

, (5)

with

x_⊥= ˜x_⊥− vˆ_⊥

1− ˜v²z

(τ0−τ), vz = τ0

τv˜z. (6)

Here,=

1− ˜vz²+(τ0/τ)²v˜²z and Fn =e^{i n}^(φ−θ)⁴F˜n

andCn correspond to thenth harmonic of the distribution function and the appropriately linearized collision kernel for Fn. A detailed description of the numerical method of solving (2) will be given in Ref. [27].

To delineate the physically interesting parameter range for our study, we first determine the range of opacities which correspond to negligible, partial or almost complete hydrodynamization. To this end, we compare at the centerr =0 of the collision, where transverse velocity is absent, the results of transport theory to the first order viscous constitutive equation P_L^{hydr o} = ^ε₃

1−¹⁶₃ _{s T}^η ₁

τ −∂rur

. With increasing system size and evolution time, fluid dynamic expectations are seen to coincide better and better with transport results, see Fig.1. The kinetic theory (2)–(4) shows hydrodynamization – that is, approximate overlap of the green and red curves in Fig.1– forτ R/γˆ. Consistent with many recent studies [23–25], this takes place prior to thermalization,PL ∼ε/3.

For the following discussion of v2/2, it is useful to rephrase the finding of Fig. 1 in terms of the properties that the collision system possesses during the typical time τ ∼ R over which the signal for v2/2 is predominantly built up in kinetic theory. Figure1 indicates then that during the timescale over which the flow is built up, the system may be characterized as not hydrodynamized forγˆ ≤ 1.5, as partially hydrodynamized for 1.5≤ ˆγ ≤4, and as almost completely hydrodynamized forγˆ ≥4. It should be under- stood that this is only a rough characterization based on the

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0 2 4 6 Opacity: γ^{^}

0 0.1 0.2 0.3 0.4 0.5

Ellipitic flow: v2/∈2

τ_s=0.15 R τs=0.55 R τ_s=0.75 R τ_s=0.95 R

Kinetic theory pre-hydrodynamic stage:

Single hit

Ideal hydro

Full transport

0 2 4 6

Opacity: γ^{^} 0

0.1 0.2 0.3 0.4 0.5

Ellipitic flow: v2/∈2

τ_s=0 τ_s=0.15 R τ_s=0.55 R τ_s=0.75 R τ_s=0.95 R

Free-streaming pre-hydrodynamic stage:

Single hit

Ideal hydro

Fig. 2 The linear response coefficientv2/2 as a function ofγˆ = R³^/⁴γ ( ε0τ0)¹^/⁴=R/l_mfp. The thick black line is the full (all orders in ˆ

γ) result obtained from evolving the kinetic theory (2), (3) up to arbi- trarily late times. The red dashed line (single hit) is the corresponding result to first order inγˆ. In the range ofγˆin which these two lines are approximately equal, the response coefficient is build up by up toO(1)

scatterings per particle. The dash-dotted lines correspond to multistage simulations where viscous fluid dynamics (withη/sset consistently withγ) is interfaced with (a) kinetic transport (left hand side) and (b) free-streaming (right hand side) at switching timeτs. The theoretical upper limit ofv2/2is obtained from evolving the initial conditions with non-viscous ideal fluid dynamic (blue dashed line)

differences between green and red curves in Fig.1, but it will be of help for discussing in the following the results of transport theory in qualitatively different dynamical regimes.

One of the main novel results of this work is the thick black curve in Fig. 2. It shows how the signal strength v2/2 calculated from the kinetic theory (2), (3) builds up smoothly over the entire physically relevant range of system sizes including systems with negligible, partial or almost complete hydrodynamization. The full solution for v2/2

approaches the analytically known [17] first order expres- sion for smallγˆ (see the ‘single-hit’ curve in Fig.2). From a technical point of view, this is a useful consistency check for the numerical accuracy of our solution. The related physics message is that in the range in which the single hit line agrees approximately with the full transport result, the signal strengthv2/2is built up from only1 collisions per particle in the kinetic theory. This is clearly consistent with our above classification of the rangeγˆ ≤ 1.5 as charac- terizing systems for which v2 is built up as a small per- turbation to free-streaming. Remarkably, Fig. 2 indicates that between one third and one half of the signal strength attained for an almost completely hydrodynamized large system ofγˆ =6 can be built up in such a much smaller non- hydrodynamized systems characterized byγˆ ≤1−1.5. This supports the qualitative idea that very small collisions, such as pp or pA, may build up a sizeable fraction of the signal strengthv2/2seen in fully hydrodynamized large collision systems while still operating close to the free-streaming limit.

4 Matching kinetic theory to viscous fluid dynamics To the extent to which collision systems hydrodynamize, one may consider describing their late-time evolution with vis-

cous fluid dynamics from a switching timeτs onwards. In the phenomenological practice of extractingη/s from data onvn, this matching of pre-hydrodynamic evolution (not nec- essarily given by full kinetic theory) is an important step in fluid dynamic models. Its uncertainty has been quantified for large collision systems which are known to hydrodynamize [28–32]. As we have seen here that smaller collision systems hydrodynamize to a lesser degree, the accuracy of this matching needs to be reassessed as a function of system size, which the calculation of the full kinetic solution allows us to do. To this end, we introduce now the viscous fluid dynamics, to which we match: We parallel the set-up of massless transport theory by considering a conformally symmetric system withε =3p. The tensor decomposition T^μν =ε

u^μu^ν +¹₃^μν

+^μνdefines the local rest frame u^μ, energy densityεand the shear viscous tensor^μν. To set the initial values of these fluid dynamic fields at the switching time τs, we match this tensor decomposition at τs to the energy-momentum tensor calculated from the distribution (4) evolved up toτswith the full kinetic theory, and with γ setting the kinetic viscosity in fluid dynamics. From time τsonwards, these fluid dynamic fields are then evolved with the Israel–Stewart viscous fluid dynamic equations

Dε+(ε+p)∇_μu^μ+_μν^μα∇_αu^ν =0, (7) (ε+p)Du^α+^αβ∇_βp+^α_ν∇_μ^μν =0, (8) τ_π,I S

D^μν+⁴₃^μν∇_αu^α

= −

^μν+2ησ^μν

. (9) Here,^μν =u^μu^ν +g^μν is the projector on the subspace orthogonal to the flow field, ∇μ is the covariant derivative and D ≡ u^μ∇μ is the comoving time derivative. Equa- tions (7) and (8) result from energy and momentum con- servation ∇_μT^μν = 0, respectively. Equation (9) ensures for a conformal system [33] that within the shear relaxation

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timeτ_π,I S, the shear viscous tensor relaxes to its Navier–

Stokes value −2ησ^μν, where η is the shear viscosity and σ^μν = ¹₂(^μα∇_αu^ν+^να∇_αu^μ)− ¹₃^μν∇_αu^α. We use the second order transport coefficientτ_π to set the Israel–

Stewart relaxation timeτπ,I S=τπ =5_sT^η =(γ ε¹^/⁴)⁻¹. In practice, we linearize [34,35] Eqs. (7)–(9) with respect to small eccentricity perturbations on top of an azimuthally symmetric background,ε = εBG+δε,u^μ =u^μ_BG+δu^μ, ^μν = ^μν_BG+δ^μν. After harmonic decomposition, this leads to a coupled set of evolution equations for 10 τ- andr-dependent fluid field components, namely four background field components and six components of second harmonic perturbations. This linearized treatment is sufficient to obtain exact results for the response coefficientsv2/2stud- ied here. The initial conditions at the switching timeτs are then evolved with a routine adapted from [34]. We calculate from the evolved fluid-dynamic fields the zeroth and second- order harmonics of the componentT^0r(τ,r)of the energy- momentum tensor, and we determinev2from the ratio of the r-integrals of these components. The values forv2 shown here are theτ → ∞ limit of this procedure. Because of the conformal symmetry, the elliptic momentum asymme- try extracted from viscous fluid dynamics can be shown to depend only on two parameters,

v2=v2(γ , τˆ s/R). (10)

5 Results from matching kinetic theory to viscous fluid dynamics

We first determine the maximal value thatv2/2can attain in a fluid-dynamic description. This maximum is obtained for an ideal fluid-dynamic evolution that translates spatial gradients into momentum gradients without dissipative losses, and that is effective over the maximal possible time, i.e. for initial conditions of (4) withτs → 0. The resulting limiting value v2/2 = 0.51 is shown as the blue dashed curve in Fig. 2.

It is substantially larger than the full kinetic theory value at ˆ

γ =6. The full transport result in Fig.2approaches this ideal fluid-dynamic upper bound slowly but steadily in the limit of very large transverse system size (γˆ → ∞). But even though we are dealing forγˆ =6 with an almost perfectly hydrody- namizing system, the system is still anisotropic and therefore, the signal strengthv2/2remains substantially reduced compared to an ideal fluid-dynamic evolution initialized at τs =0.

From the dash-dotted curves in Fig.2a, one sees that viscous fluid dynamics, matched to the pre-hydrodynamic evolution atτs, approaches the full kinetic theory calculation of v2/2smoothly for increasingτs. So for fixedγˆ,v2/2starts to quantitatively agree with full transport for τs R/γˆ, consistent with the observation in Fig.1that the constitutive equations are approximately fulfilled. For earlier initializa-

tions of fluid dynamics, sayτs <R/2γˆ, the signalv2/2is too strong. Indeed, for too early times,P_L^{hydr o}turns negative, signaling a catastrophic failure of fluid dynamics, see Fig.1.

However, whether the matching of kinetic theory to viscous fluid dynamics is a quantitatively satisfactory approximation to full kinetic theory depends on the accuracy that one wants to achieve. The phenomenological challenge is to determine γˆ for fixedv2/2, a task that becomes more challenging for large γˆ where the γˆ-dependence of v2/2

becomes weak. For instance, for a fixed valuev2/2≈0.32, a simulation using matching at τs = 0.15R would yield

ˆ

γ ≈ 2 while the truth of the full transport calculation is ˆ

γ ≈4. This illustrates that even small uncertainties inv2/2

for fixedγˆ can result in large uncertainties in extracting γˆ from a givenv2/2. As η/sis inversely proportional to γˆ, this poses a challenge for extractingη/sfrom fluid dynamic simulation with accuracy significantly better than a factor of two.

While we have discussed so far only the use of full kinetic theory for the pre-hydrodynamic stage up to τs, a much more approximate, simplified procedure is currently in phenomenological use. It consists of initializing fluid dynamics from free-streamed distributions at timeτs [29].

This may be justified qualitatively on the grounds that both free-streaming and kinetic transport smoothen gradients in initial distributions and that any difference between free- streaming and transport will emerge only gradually at times τ ∼ lmfp ∼ R/γˆ at which fluid dynamics starts to give a good description of theT^μν-evolution (see Fig.1). Figure2b shows that matching viscous fluid dynamics to free-streamed initial distributions comes with large uncertainties displayed by the wide spread of curves for differentτs.

6 In summary

we have provided a full kinetic theory calculation of the opacity dependence of elliptic flow, ranging from systems that are sufficiently small to evolve close to free-streaming, up to systems that are sufficiently large to exhibit fluid dynamic behavior already at times τ R. We find in very small systems a surprisingly rapid onset of signal strength v2/2

with system size. In particular, very small collision systems that allow for only up to one isotropizing large-angle scat- tering per particle excitation and that do not hydrodynamize significantly on time scalesτ <2Rare still found to build up one third to one half of the signal strength observed in almost completely hydrodynamized, large collision systems.

Thatv2/2rises with system size most rapidly in the range up to R < lmfp where collective flow results from perturbative (inγˆ) corrections to free-streaming is a characteristic feature of kinetic transport theory established here. These findings are qualitatively in line with the potentially contra-

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dictory requirements thatv2attains sizeable values already in the smallestppcollision systems despite many other obser- vations in pp collisions being seemingly consistent with an approximate free-streaming picture.

Our study demonstrates that matching a kinetic theory pre-hydrodynamic stage at τs to a viscous fluid dynamic description can yield accurate results forv2/2if the switching timeτsis sufficiently late and if the system is sufficiently opaque (γˆ 1). The accuracy of this matching degrades only gradually with decreasingγˆ. We note that for systems for which the pre-equilibration dynamics is sufficiently short, full kinetic transport may be replaced by linear response [31,32,36], for which a numerical code KøMPøST [31,32]

is available. However, in smaller systems, for which the pre- equilibirum dynamics needs to be followed to later times τs ∼R, full kinetic transport is needed, and for even smaller systemsγ <ˆ 1 the single hit approximation to kinetic theory is sufficient.

In short, we have demonstrated that while pp collisions are very different from AA collisions since they realize a close to free-streaming picture that differs qualitatively from hydrodynamics, the collectivity in both systems can still arise from the same microscopic interactions.

Data Availability StatementThis manuscript has no associated data or the data will not be deposited. [Authors’ comment: All results shown in the figures follow directly and unambiguously from the formulas displayed in the manuscript.]

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP³.

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